Arcs and tensors

Abstract

To an arc $\mathcal{A}$ of $\mathrm{PG}(k-1,q)$ of size $q+k-1-t$ we associate a tensor in $⟨{\nu }_{k,t}(\mathcal{A}){⟩}^{\otimes k-1}$, where ${\nu }_{k,t}$ denotes the Veronese map of degree $t$ defined on $\mathrm{PG}(k-1,q)$. As a corollary we prove that for each arc $\mathcal{A}$ in $\mathrm{PG}(k-1,q)$ of size $q+k-1-t$, which is not contained in a hypersurface of degree $t$, there exists a polynomial $F(Y_1,\dots ,Y_{k-1})$ (in $k(k-1)$ variables) where $Y_j=(X_{j1},\dots ,X_{jk})$, which is homogeneous of degree $t$ in each of the $k$-tuples of variables $Y_j$, which upon evaluation at any $(k-2)$-subset $S$ of the arc $\mathcal{A}$ gives a form of degree $t$ on $\mathrm{PG}(k-1,q)$ whose zero locus is the tangent hypersurface of $\mathcal{A}$ at $S$, i.e. the union of the tangent hyperplanes of $\mathcal{A}$ at $S$. This generalises the equivalent result for planar arcs ($k=3$), proven in \cite{BaLa2018}, to arcs in projective spaces of arbitrary dimension. A slightly weaker result is obtained for arcs in $\mathrm{PG}(k-1,q)$ of size $q+k-1-t$ which are contained in a hypersurface of degree $t$. We also include a new proof of the Segre-Blokhuis-Bruen-Thas hypersurface associated to an arc of hyperplanes in $\mathrm{PG}(k-1,q)$.